Some examples of classical coboundary Lie bialgebras with coboundary duals

Some examples are given of finite dimensional Lie bialgebras whose brackets and cobrackets are determined by pairs of r-matrices. The aim of this Letter is to give some low-dimensional examples of classical coboundary Lie bialgebras [1, 2] with coboundary duals. Since such structures can be specified (up to automorphisms) by pairs of r-matrices, so it is natural to call them bi-r-matrix bialgebras (BrB). There are some reasons to study BrB. Assuming that both Lie bialgebras of a dual pair are coboundary, we impose additional constrains, which can facilitate the search of new classical r-matrices connected with nonsemisimple Lie algebras. Recall that many Lie algebras of physical interest are nonsemisimple ones, but up to now there is detailed classification of r-matrices only for the complex simple Lie algebras [3]. The presence of a pair of r-matrices is useful for practical quantization of Lie bialgebras [1] permitting more symmetrical treatment of quantum algebras and groups. However, most interesting applications of BrB are possible in the theory of bihamiltonian dynamical systems [4, 5]. In this case the presence of a pair of r-matrices allows us to define the pair of dynamical systems on the space which is the space of the original Lie algebra canonically identified with its dual space [6]. Now recall some basic definitions [1, 2]. Let L be a finite-dimensional Lie algebra and L the dual space of L in respect to a nondegenerate bilinear form < ., . > on L × L. The Lie algebra L is called a bialgebra, if there exist a map δ : L → L⊗ L which is an 1-cocycle δ([x, y]) = [δ(x), y ⊗ 1 + 1⊗ y] + [x⊗ 1 + 1⊗ x, δ(y)], x, y ∈ L (1) and which defines on L a Lie algebra structure [., .]∗ : L ∗ ⊗ L → L by the following relation < [ξ, η]∗, x >=< ξ ⊗ η, δ(x) >, x ∈ L, ξ, η ∈ L ∗. (2) If one puts δ(x) = [x⊗ 1 + 1⊗ x, r], r = rem ⊗ en ∈ L⊗ L (3) then the 1-cocycle condition (1) is fulfilled identically. In this case the Lie bialgebra L is called a coboundary one. It can be shown [2] that the Jacobi identity for the elements of L is equivalent to the following equation [x⊗ 1⊗ 1 + 1⊗ x⊗ 1 + 1⊗ 1⊗ x, [r, r]s] = 0, x ∈ L. (4)